where \(A\)  and \(B\)  are the Hamaker constants corresponding to the attractive and repulsive contributions, respectively. For graphitic systems, one typically takes the values for graphite \(A = 15.2\)  eV\(Å^6\)  and \(B = 24\times 10^3\)  eV\(Å^{12}\)  [1]}. \(\sigma = 4/\sqrt{3}a^2\)  is the mean surface density of Carbon atoms with \(a = 2.49\)  \(Å\)  being the lattice constant. Also, the distance between the CNT surfaces is \(\rho \) . Perfroming the integration over the length of the two CNTs with radii \(R_{1}\)  and \(R_{2}\) , the LJ-vdW interaction can be written as [2]}:
\(V_{vdW} &=& -\frac{3\pi A\sigma ^{2}R_{1}R_{2}}{8}\int _{0}^{2\pi }\int _{0}^{2\pi }\frac{1}{r^5}d\varphi _{1}d\varphi _{2}\nonumber \\& & + \frac{63\pi B\sigma ^{2}R_{1}R_{2}}{256}\int _{0}^{2\pi }\int _{0}^{2\pi }\frac{1}{r^{11}}d\varphi _{1}d\varphi _{2},\) 
